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Thread starter 3 weeks ago
#1
Question:

Under a certain transformation the image of the point (x,y) is (X,Y) where . This transformation maps any point on the line onto another point on the line . Find the two possible values of m.

My attempt:
I tried to look at the the effect of the transformation on the unit square

Since it says y= mx
i thought it had something to do with reflection in the line y=mx where
I have tried to calculate the angle between i.e ( 1,0) and its image (1,2) or (0,1) (4,3) but it could not figure it out.

I would like to know if my approach is wrong or right?
Please can i have help on finding the two values of m
Last edited by bigmansouf; 3 weeks ago
0
3 weeks ago
#2
(Original post by bigmansouf)
Question:

Under a certain transformation the image of the point (x,y) is (X,Y) where . The transformation maps any point on the line . Find the two possible values of m.

My attempt:
I tried to look at the the effect of the transformation on the unit square

Since it says y= mx
i thought it had something to do with reflection in the line y=mx where
I have tried to calculate the angle between i.e ( 1,0) and its image (1,2) or (0,1) (4,3) but it could not figure it out.

I would like to know if my approach is wrong or right?
Please can i have help on finding the two values of m
Just evaluate

Though the question doesn't really make sense. It doesn't give any conditions to work with. Is it meant to say that every point on this line maps to itself?
1
Thread starter 3 weeks ago
#3
(Original post by RDKGames)
Just evaluate

Though the question doesn't really make sense. It doesn't give any conditions to work with. Is it meant to say that every point on this line maps to itself?
I edited it to include the part i missed

'This transformation maps any point on the line y = mx onto another point on the line y=mx'
0
3 weeks ago
#4
(Original post by bigmansouf)
I edited it to include the part i missed

'This transformation maps any point on the line y = mx onto another point on the line y=mx'
Cool, so just use the fact that (Y,X) lies on y=mx... so Y=mX must hold.
0
3 weeks ago
#5
(Original post by bigmansouf)
Question:

Under a certain transformation the image of the point (x,y) is (X,Y) where . This transformation maps any point on the line onto another point on the line . Find the two possible values of m.

My attempt:
I tried to look at the the effect of the transformation on the unit square

Since it says y= mx
i thought it had something to do with reflection in the line y=mx where
I have tried to calculate the angle between i.e ( 1,0) and its image (1,2) or (0,1) (4,3) but it could not figure it out.

I would like to know if my approach is wrong or right?
Please can i have help on finding the two values of m
Isn't this equivalent to finding the eigenvectors of the matrix?

Because then if v is an eigenvector of M, Mv = kv, i.e. the transformation under M is colinear to the vector you started with, which is essentially what this mapping line to line business is saying. If you're not sure how to calculate eigenvectors of a matrix, there is a wealth of information line describing this process.

Spoiler:
Show
The eigenvectors are (1,1)^T, (-2,1)^T so the lines would be y = x and y = -1/2 * x (I think)
Last edited by Ryanzmw; 3 weeks ago
0
Thread starter 3 weeks ago
#6

I learnt that the basics of egienvectors is

therefore

I sub each into the matrix;

when

thus y=x

when

thus

Did i follow the right approach?

(Original post by Ryanzmw)
Isn't this equivalent to finding the eigenvectors of the matrix?

Because then if v is an eigenvector of M, Mv = kv, i.e. the transformation under M is colinear to the vector you started with, which is essentially what this mapping line to line business is saying. If you're not sure how to calculate eigenvectors of a matrix, there is a wealth of information line describing this process.

Spoiler:
Show

The eigenvectors are (1,1)^T, (-2,1)^T so the lines would be y = x and y = -1/2 * x (I think)
Last edited by bigmansouf; 3 weeks ago
1
3 weeks ago
#7
Yep. Do you see how this relates to the question?
(Original post by bigmansouf)

I learnt that the basics of egienvectors is

therefore

I sub each into the matrix;

when

thus y=x

when

thus

Did i follow the right approach?
2
3 weeks ago
#8
And the result in post#1 shows one of these:

the third column is transformed from column(1,1) to column(5,5) ... ie a point on the line y=x, (1,1) gets transformed to another point on the line y=x, (5,5) ...
1
Thread starter 3 weeks ago
#9
yes i do thank you
(Original post by Ryanzmw)
Yep. Do you see how this relates to the question?
0
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